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In a previous work (M. Campisi. Stud. Hist. Phil. M. P. 36 (2005) 275-290) wehave addressed the mechanical foundations of equilibrium thermodynamics on thebasis of the Generalized Helmholtz Theorem. It was found that the volumeentropy provides a good mechanical analogue of thermodynamic entropy because itsatisfies the heat theorem and it is an adiabatic invariant. This propertyexplains the ``equal'' sign in Clausius principle ($S_f \geq S_i$) in a purelymechanical way and suggests that the volume entropy might explain the ``largerthan'' sign (i.e. the Law of Entropy Increase) if non adiabatic transformationswere considered. Based on the principles of microscopic (quantum or classical)mechanics here we prove that, provided the initial equilibrium satisfy thenatural condition of decreasing ordering of probabilities, the expectationvalue of the volume entropy cannot decrease for arbitrary transformationsperformed by some external sources of work on a insulated system. This can beregarded as a rigorous quantum mechanical proof of the Second Law. We discusshow this result relates to the Minimal Work Principle and improves overprevious attempts. The natural evolution of entropy is towards larger valuesbecause the natural state of matter is at positive temperature. Actually theLaw of Entropy Decrease holds in artificially prepared negative temperaturesystems.

Mechanical Proof of the Second Law of Thermodynamics Based on Volume Entropy